Triangles

1. Triangles

2. Information

3. Triangles A triangle is a polygon made up of three connected line segments that form a closed figure. Triangles are named using the triangle symbol, △, and its vertices, e.g., △ABC. exterior AB, BC and AC are the line segments that make up the triangle. interior The interior is the set of points inside the triangle. The exterior is the set of points outside the triangle.

4. Classifying triangles by angle What is special about the angles in each of the triangles? right triangle acute triangle one right angle three acute angles obtuse triangle equiangular triangle one obtuse angle three congruent angles

5. Classifying triangles by side length What is special about the sides of each triangle? isosceles triangle equilateral triangle scalene triangle no congruent sides three congruent sides two congruent sides

6. Classifying triangles

7. Angle relationships in triangles

8. Triangle sum theorem Triangle sum theorem: all three angle measures in a triangle add up to 180°. m∠1 + m∠2 + m∠3 = 180° Prove the triangle sum theorem. alternate interior angle theorem: m∠1 = m∠4 and m∠3=m∠5 angle addition postulate: m∠4 + m∠2 + m∠5 = 180° angles along a line sum to 180° substituting: since m∠1 = m∠4 and m∠3=m∠5  m∠1 + m∠2 + m∠3 = 180°

9. Using the triangle sum theorem What are the angle measures in an equiangular triangle? The measures of the angles in a triangle add up to 180°. An equiangular triangle has three congruent angles. equiangular triangle: x° = m∠A = m∠B = m∠C triangle sum theorem: m∠A + m∠B + m∠C = 180° x° + x° + x° = 3x° = 180° substituting: solving: x° = 180° ÷ 3 = 60° What is the sum of the measures on the non-right angles in a right triangle? triangle sum theorem: m∠A + m∠B + m∠C = 180° substituting: m∠A + m∠B + 90° = 180° solving: m∠A + m∠B = 180 ° – 90° = 90° They are complementary angles.

10. Exterior angle theorem

11. Third angles theorem